Knot knock

August 5, 2008 at 8:16 am | Posted in Math Learning | Leave a comment

Knot table to Eight crossing from Lickorish,1997

Do you know how hard to draw the knot picture?!…
You must try it!!
Look at the eight crossings knot table…

  • First, You should know how to tie a knot, thus you can know the upper segment and lower part!
  • Second, Upper segment is a real black line, and the lower part must anticipate an empty.
    ※You should keep all basic knot form deeply in your brain.
  • Third, Draw it with your pen. 3-dimension memory and 2-dimension drawing. Keep practice, practice and practice or you cannot draw it as well.(Actually speaking I still cannot do it well now)
  • Fourth, Reverse what you draw…to show the different positive picture and negative picture like.

This is What I did before…A never succeed work><

After going to Taipei, I open on the book: W.B. Raymond Lickorish. An Introduction to Knot Theory, GTM;175. Springer-Verlag New York(1997).To turn on my memory of Knot…It was Chun Chung tell me the good one. Seems a hard reading book for a undergraduate guy…I always confuse that Why I can easily understand what knot is after Chun Chung easily explained it, but cannot understand what the book show us. Alas~O if there I can take part in a course of he to tell me what the knot is…Not just a book.

Alwxander Polynomial Table from Lickrish,1997,p.59

Take something for example, look at the definition1.1 to 1.4, and compare with knot table to Eight Crossing(table one). To make the definition clearly had been a hard work for me…

Definition1.1: A link L of m components is a subset of S3 or of R3, that consists of m disjoint, piecewise linear, simple closed curves. A link of one component is a Knot.

Definition1.2: Links L1 and L2 in S3 are equivalent if there is an orientation-preserving piecewise linear homeomorphism H: S3 → S3 such that h(L1)=(L2)

Note Definition: A knot is said to be the unknot if it bounds an embedded piecewise linear discs in S3.

Definition1.3: A knot K is a prime knot if it is not the unknot, and K=K1+K2 implies that K1 or K2 is the unknot.

Definition1.4 Suppose that L is a two-component oriented link with components L1 and L2. The linking number lk(L,L) of L1 and L2 is half the sum of the sign, in a diagram for L, of the crossings at which one strand is from L1 and the other is from L2

My learning problems always stubs my brain..

Here: Who knows the “link”?! In mathworld shows that ” Formally, a link is one or more disjointly embedded circles in three space. More informally, a link is an assembly of knots with mutual entaglements”( The definitions of knot and link are much easier on the page with picture( course, I understand it now, but it quite hard for me to state the definition. Cause I found at least 3 definition of Knot or link…

And: Is there anybody can show me the knot that been bounded an embedded piecewise linear discs in S3 (in definition1.2) ?….It sounds like noodles in a big pan. God~There must no people read this book like me…@@

OK~take it easy: Who knows the number of Prime knots after see the definition? We can distinguish the prime knot from definion1.3.. also can know the number of Prime knots not equal to the prime number from the knot table. As for the definition1.4…How does the “half the sum of the sign” work?…It just says that something related in homology theory. How can I catch the meanings of it from such few words?!

After looking over these, how you think about to know other important invariant-related topics like Alexander polynomial, Jones polynomial and so on in knot theory? It should be another hard work to overcome.  Of course, you can see what the stupid student’s story about learning math, especially in complex geometry , Lie algebra, knots and those with Quantization on the blog from now on.  Thus for the important part of knot,Alexander polynomial and Jones polynomial, you will see what the  learning problem I met.

Jones Polynomial Table from Lickorish,1997


Practice drawing between dark and bright side

July 14, 2008 at 4:43 am | Posted in Others | Leave a comment

brick drawing on Jun 19th

brick drawing on Jun 19th

It takes me almost 4 hours to draw the brick. My uncle told me must use the exquisite eyeside to see the bright side and shadow…try to use the pencil with many different ways. And then, just like the drawing finished on June 19th, you may see on the post .Gee learned a lot during these hard days… Not only for drawing pictures, but finding my beating heart, It is a life process.

Maybe I will type the thought process changed and grew when I talked to more wonderful guy… “Thank you, all.” must be some words I would like to appreciate my family, friends and teachers. Though there is still a depression-days for me, remember that the light behide the shadow. You can choose to turn over with free action, except that the fixed cage never changed.

My oops translate

April 16, 2008 at 1:34 am | Posted in Math Learning | 1 Comment
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OOPS, which means “Opensource Opencourseware Prototype System”, is a wonderful group that  translate some opensources for Chinese. Here is the page(

Gee took part in this group monthes ago, to do further education and do something great for people. Almost all course I chsosed here did come from MIT courses. The topics I interested in are just like the following:

Of course, some basical course are needed here…

Keep growing, keep working, and keep translating…it will be a good beginning. Though it still a life long road after, never afraid of that…just keep going to do the right things…I am telling myself.

Now I am searching for the couse string and knots.Maybe I will share what I study more in my mind from the course here after…

Sent the picture to Fei-Hung Ho

March 5, 2008 at 2:15 am | Posted in Others | 1 Comment

A picture for Fei-Hung Ho…

Position: Teaching Building 7F

2008 Taitung International School/Workshop on Cosmology and Gravitation

1st write

February 28, 2008 at 9:09 am | Posted in Others | 1 Comment

  It’s my first time to use the new blog of….For learning and making my dull brains become a clear one.

   Remember John Dewey. Always know “knowledge is power”(Sir Francis Bacon). Besure a wisdom person…Try to Make yourself stronger~~Your soul must needs springs, your body always needs foods. Touching the beating heart… what life be going on and on.

   Believe that nature will give you the answer result.

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